For the sake of completeness, we prove this result along similar line as in Errais et al. [1]}.
For a fixed time \(t\) , \((M_t, S_t, \lambda _t, \mu _t)\)  has right-continuous paths of finite variation.
Hence,
\(f (M_t, S_t, \lambda _t, \mu _t) - f (M_0, S_0, \lambda _0, \mu _0)&=&\int _0^t r (\lambda ^* - \lambda _u) \frac{\partial f(M_u, S_u, \lambda _u, \mu _u)}{\partial \lambda } \ du\nonumber \\&&+ \int _0^t s (\mu ^* - \mu _u) \frac{\partial f(M_u, S_u, \lambda _u, \mu _u)}{\partial \mu }\ du\\&&+ \sum _{0 < u \le t} \left[ f(M_u, S_u, \lambda _u, \mu _u) - f(M_{u-}, S_{u-}, \lambda _{u-}, \mu _{u-}) \right].\nonumber \) 
